Quantum Mechanics, Volume 3 E-Book

Claude Cohen-Tannoudji

• Herausgeber: Wiley-VCH Verlag GmbH & Co. KGaA
• Kategorie: Wissenschaft und neue Technologien
• Sprache: Englisch

Seitenzahl: 1472

Veröffentlichungsjahr: 2020

Beschreibung

This new, third volume of Cohen-Tannoudji's groundbreaking textbook covers advanced topics of quantum mechanics such as uncorrelated and correlated identical particles, the quantum theory of the electromagnetic field, absorption, emission and scattering of photons by atoms, and quantum entanglement. Written in a didactically unrivalled manner, the textbook explains the fundamental concepts in seven chapters which are elaborated in accompanying complements that provide more detailed discussions, examples and applications.

* Completing the success story: the third and final volume of the quantum mechanics textbook written by 1997 Nobel laureate Claude Cohen-Tannoudji and his colleagues Bernard Diu and Franck Laloë
* As easily comprehensible as possible: all steps of the physical background and its mathematical representation are spelled out explicitly
* Comprehensive: in addition to the fundamentals themselves, the books comes with a wealth of elaborately explained examples and applications

Claude Cohen-Tannoudji was a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris where he also studied and received his PhD in 1962. In 1973 he became Professor of atomic and molecular physics at the Collège des France. His main research interests were optical pumping, quantum optics and atom-photon interactions. In 1997, Claude Cohen-Tannoudji, together with Steven Chu and William D. Phillips, was awarded the Nobel Prize in Physics for his research on laser cooling and trapping of neutral atoms.

Bernard Diu was Professor at the Denis Diderot University (Paris VII). He was engaged in research at the Laboratory of Theoretical Physics and High Energy where his focus was on strong interactions physics and statistical mechanics.

Franck Laloë was a researcher at the Kastler-Brossel laboratory of the Ecole Normale Supérieure in Paris. His first assignment was with the University of Paris VI before he was appointed to the CNRS, the French National Research Center. His research was focused on optical pumping, statistical mechanics of quantum gases, musical acoustics and the foundations of quantum mechanics.

Leseprobe

Table of Contents

Cover

Foreword

Chapter XV: Creation and annihilation operators for identical particles

A. General formalism

B. One-particle symmetric operators

C. Two-particle operators

COMPLEMENTS OF CHAPTER XV, READER’S GUIDE

Complement A

XV

Particles and holes

1. Ground state of a non-interacting fermion gas

2. New definition for the creation and annihilation operators

3. Vacuum excitations

Complement B

XV

Ideal gas in thermal equilibrium; quantum distribution functions

1. Grand canonical description of a system without interactions

2. Average values of symmetric one-particle operators

3. Two-particle operators

4. Total number of particles

5. Equation of state, pressure

Complement C

XV

Condensed boson system, Gross-Pitaevskii equation

1. Notation, variational ket

2. First approach

3. Generalization, Dirac notation

4. Physical discussion

Complement D

XV

Time-dependent Gross-Pitaevskii equation

1. Time evolution

2. Hydrodynamic analogy

3. Metastable currents, superfluidity

Complement E

XV

Fermion system, Hartree-Fock approximation

1. Foundation of the method

2. Generalization: operator method

Complement F

XV

Fermions, time-dependent Hartree-Fock approximation

1. Variational ket and notation

2. Variational method

3. Computing the optimizer

4. Equations of motion

Complement G

XV

Fermions or Bosons: Mean field thermal equilibrium

1. Variational principle

2. Approximation for the equilibrium density operator

3. Temperature dependent mean field equations

Complement H

XV

Applications of the mean field method for non-zero temperature (fermions and bosons)

1. Hartree-Fock for non-zero temperature, a brief review

2. Homogeneous system

3. Spontaneous magnetism of repulsive fermions

4. Bosons: equation of state, attractive instability

Chapter: XVI Field operator

A. Definition of the field operator

B. Symmetric operators

C. Time evolution of the field operator (Heisenberg picture)

D. Relation to field quantization

COMPLEMENTS OF CHAPTER XVI, READER’S GUIDE

Complement A

XVI

Spatial correlations in an ideal gas of bosons or fermions

1. System in a Fock state

2. Fermions in the ground state

3. Bosons in a Fock state

Complement B

XVI

Spatio-temporal correlation functions, Green’s functions

1. Green’s functions in ordinary space

2. Fourier transforms

3. Spectral function, sum rule

Complement C

XVI

Wick’s theorem

1. Demonstration of the theorem

2. Applications: correlation functions for an ideal gas

Chapter: XVII Paired states particles of identical

A. Creation and annihilation operators of a pair of particles

B. Building paired states

C. Properties of the kets characterizing the paired states

D. Correlations between particles, pair wave function

E. Paired states as a quasi-particle vacuum; Bogolubov-Valatin transformations

COMPLEMENTS OF CHAPTER XVII, READER’S GUIDE

Complement A

XVII

Pair field operator for identical particles

1. Pair creation and annihilation operators

2. Average values in a paired state

3. Commutation relations of field operators

Complement B

XVII

Average energy in a paired state

1. Using states that are not eigenstates of the total particle number

2. Hamiltonian

3. Spin 1/2 fermions in a singlet state

4. Spinless bosons

Complement C

XVII

Fermion pairing, BCS theory

1. Optimization of the energy

2. Distribution functions, correlations

3. Physical discussion

4. Excited states

Complement D

XVII

Cooper pairs

1. Cooper model

2. State vector and Hamiltonian

3. Solution of the eigenvalue equation

4. Calculation of the binding energy for a simple case

Complement E

XVII

Condensed repulsive bosons

1. Variational state, energy

2. Optimization

3. Properties of the ground state

4. Bogolubov operator method

Chapter: XVIII REVIEW OF CLASSICAL ELECTRODYNAMICS

A. Classical electrodynamics

B. Describing the transverse field as an ensemble of harmonic oscillators

COMPLEMENTS OF CHAPTER XVIII, READER’S GUIDE

Complement A

XVIII

Lagrangian formulation of electrodynamics

1. Lagrangian with several types of variables

2. Application to the free radiation field

3. Lagrangian of the global system field + interacting particles

Chapter: XIX QUANTIZATION OF ELECTROMAGNETIC RADIATION

A. Quantization of the radiation in the Coulomb gauge

B. Photons, elementary excitations of the free quantum field

C. Description of the interactions

COMPLEMENTS OF CHAPTER XIX, READER’S GUIDE

Complement A

XIX

Momentum exchange between atoms and photons

1. Recoil of a free atom absorbing or emitting a photon

2. Applications of the radiation pressure force: slowing and cooling atoms

3. Blocking recoil through spatial confinement

4. Recoil suppression in certain multi-photon processes

Complement B

XIX

Angular momentum of radiation

1. Quantum average value of angular momentum for a spin 1 particle

2. Angular momentum of free classical radiation as a function of normal variables2047

3. Discussion

Complement C

XIX

Angular momentum exchange between atoms and photons

1. Transferring spin angular momentum to internal atomic variables

2. Optical methods

3. Transferring orbital angular momentum to external atomic variables

Chapter: XX ABSORPTION, EMISSION AND SCATTERING OF PHOTONS BY ATOMS

A. A basic tool: the evolution operator

B. Photon absorption between two discrete atomic levels

C. Stimulated and spontaneous emissions

D. Role of correlation functions in one-photon processes

E. Photon scattering by an atom

COMPLEMENTS OF CHAPTER XX, READER’S GUIDE

Complement A

XX

A multiphoton process: two-photon absorption

1. Monochromatic radiation

2. Non-monochromatic radiation

3. Discussion

Complement B

XX

Photoionization

1. Brief review of the photoelectric effect

2. Computation of photoionization rates

3. Is a quantum treatment of radiation necessary to describe photoionization? .

4. Two-photon photoionization

5. Tunnel ionization by intense laser fields

Complement C

XX

Two-level atom in a monochromatic field. Dressed-atom method

1. Brief description of the dressed-atom method

2. Weak coupling domain

3. Strong coupling domain

4. Modifications of the field. Dispersion and absorption

Complement D

XX

Light shifts: a tool for manipulating atoms and fields

1. Dipole forces and laser trapping

2. Mirrors for atoms

3. Optical lattices

4. Sub-Doppler cooling. Sisyphus effect

5. Non-destructive detection of a photon

Complement E

XX

Detection of one- or two-photon wave packets, interference

1. One-photon wave packet, photodetection probability

2. One- or two-photon interference signals

3. Absorption amplitude of a photon by an atom

4. Scattering of a wave packet

5. Example of wave packets with two entangled photons

Chapter: XXI QUANTUM ENTANGLEMENT, MEASUREMENTS, BELL’S INEQUALITIES

A. Introducing entanglement, goals of this chapter

B. Entangled states of two spin-1/2 systems

C. Entanglement between more general systems

D. Ideal measurement and entangled states

E. “Which path” experiment: can one determine the path followed by the photon in Young’s double slit experiment?

F. Entanglement, non-locality, Bell’s theorem

COMPLEMENTS OF CHAPTER XXI, READER’S GUIDE

Complement A

XXI

Density operator and correlations; separability

1. Von Neumann statistical entropy

2. Differences between classical and quantum correlations

3. Separability

Complement B

XXI

GHZ states, entanglement swapping

1. Sign contradiction in a GHZ state

2. Entanglement swapping

Complement C

XXI

Measurement induced relative phase between two condensates

1. Probabilities of single, double, etc. position measurements

2. Measurement induced enhancement of entanglement

3. Detection of a large number Q of particles

Complement D

XXI

Emergence of a relative phase with spin condensates; macroscopic non-locality and the EPR argument

1. Two condensates with spins

2. Probabilities of the different measurement results

3. Discussion

Appendix IV: Feynman path integral

1. Quantum propagator of a particle

2. Interpretation in terms of classical histories

3. Discussion; a new quantization rule

4. Operators

Appendix V: Lagrange multipliers

1. Function of two variables

2. Function of AT variables

Appendix VI: Brief review of Quantum Statistical Mechanics

1. Statistical ensembles

2. Intensive or extensive physical quantities

Appendix VII: Wigner transform

1. Delta function of an operator

2. Wigner distribution of the density operator (spinless particle)

3. Wigner transform of an operator

4. Generalizations

5. Discussion: Wigner distribution and quantum effects

Bibliography of volume III

Index

End User License Agreement

Guide

Cover

Table of Contents

Begin Reading

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QUANTUM MECHANICS Volume III

Fermions, Bosons, Photons, Correlations, and Entanglement

Claude Cohen-Tannoudji

Bernard Diu

Franck Laloë

Translated from the French by Nicole Ostrowsky and Dan Ostrowsky

Authors

First Edition

Prof. Dr. Claude Cohen-Tannoudji

Laboratoire Kastler Brossel (ENS)

24 rue Lhomond

75231 Paris Cedex 05

France

Prof. Dr. Bernard Diu

4 rue du Docteur Roux

91440 Boures-sur-Yvette

France

Prof. Dr. Frank Laloe

Laboratoire Kastler Brossel (ENS)

24 rue Lhomond

75231 Paris Cedex 05

France

Cover Image

© antishock/Getty Images

First Edition

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Directions for Use

This book is composed of chapters and their complements:

–

The chapters

contain the fundamental concepts. Except for a few additions and variations, they correspond to a course given in the last year of a typical undergraduate physics program (Volume I) or of a graduate program (Volumes II and III). The 21 chapters are

complete in themselves

and can be studied independently of the complements.

–

The complements

follow the corresponding chapter. Each is labelled by a letter followed by a subscript, which gives the number of the chapter (for example, the complements of Chapter V are, in order, A

V

, B

V

, C

V

, etc.). They can be recognized immediately by the symbol • that appears at the top of each of their pages.

The complements vary in character. Some are intended to expand the treatment of the corresponding chapter or to provide a more detailed discussion of certain points. Others describe concrete examples or introduce various physical concepts. One of the complements (usually the last one) is a collection of exercises.

The difficulty of the complements varies. Some are very simple examples or extensions of the chapter. Others are more difficult and at the graduate level or close to current research. In any case, the reader should have studied the material in the chapter before using the complements.

The complements are generally independent of one another. The student should not try to study all the complements of a chapter at once. In accordance with his/her aims and interests, he/she should choose a small number of them (two or three, for example), plus a few exercises. The other complements can be left for later study. To help with the choise, the complements are listed at the end of each chapter in a “reader’s guide”, which discusses the difficulty and importance of each.

Some passages within the book have been set in small type, and these can be omitted on a first reading.

Foreword

Quantum mechanics is a branch of physics whose importance has continually increased over the last decades. It is essential for understanding the structure and dynamics of microscopic objects such as atoms, molecules and their interactions with electromagnetic radiation. It is also the basis for understanding the functioning of numerous new systems with countless practical applications. This includes lasers (in communications, medicine, milling, etc.), atomic clocks (essential in particular for the GPS), transistors (communications, computers), magnetic resonance imaging, energy production (solar panels, nuclear reactors), etc. Quantum mechanics also permits understanding surprising physical properties such as superfluidity or supraconductivity. There is currently a great interest in entangled quantum states whose non-intuitive properties of nonlocality and nonseparability permit conceiving remarkable applications in the emerging field of quantum information. Our civilization is increasingly impacted by technological applications based on quantum concepts. This why a particular effort should be made in the teaching of quantum mechanics, which is the object of these three volumes.

The first contact with quantum mechanics can be disconcerting. Our work grew out of the authors’ experiences while teaching quantum mechanics for many years. It was conceived with the objective of easing a first approach, and then aiding the reader to progress to a more advance level of quantum mechanics. The first two volumes, first published more than forty years ago, have been used throughout the world. They remain however at an intermediate level. They have now been completed with a third volume treating more advanced subjects. Throughout we have used a progressive approach to problems, where no difficulty goes untreated and each aspect of the diverse questions is discussed in detail (often starting with a classical review).

This willingness to go further “without cheating or taking shortcuts” is built into the book structure, using two distinct linked texts: chapters and complements. As we just outlined in the “Directions for use”, the chapters present the general ideas and basic concepts, whereas the complements illustrate both the methods and concepts just exposed.

Volume I presents a general introduction of the subject, followed by a second chapter describing the basic mathematical tools used in quantum mechanics. While this chapter can appear long and dense, the teaching experience of the authors has shown that such a presentation is the most efficient. In the third chapter the postulates are announced and illustrated in many of the complements. We then go on to certain important applications of quantum mechanics, such as the harmonic oscillator, which lead to numerous applications (molecular vibrations, phonons, etc.). Many of these are the object of specific complements.

Volume II pursues this development, while expanding its scope at a slightly higher level. It treats collision theory, spin, addition of angular momenta, and both time-dependent and time-independent perturbation theory. It also presents a first approach to the study of identical particles. In this volume as in the previous one, each theoretical concept is immediately illustrated by diverse applications presented in the complements. Both volumes I and II have benefited from several recent corrections, but there have also been additions. Chapter XIII now contains two sections §§ D and E that treat random perturbations, and a complement concerning relaxation has been added.

Volume III extends the two volumes at a slightly higher level. It is based on the use of the creation and annihilation operator formalism (second quantization), which is commonly used in quantum field theory. We start with a study of systems of identical particles, fermions or bosons. The properties of ideal gases in thermal equilibrium are presented. For fermions, the Hartree-Fock method is developed in detail. It is the base of many studies in chemistry, atomic physics and solid state physics, etc. For bosons, the Gross-Pitaevskii equation and the Bogolubov theory are discussed. An original presentation that treats the pairing effect of both fermions and bosons permits obtaining the BCS (Bardeen-Cooper-Schrieffer) and Bogolubov theories in a unified framework. The second part of volume III treats quantum electrodynamics, its general introduction, the study of interactions between atoms and photons, and various applications (spontaneous emission, multiphoton transitions, optical pumping, etc.). The dressed atom method is presented and illustrated for concrete cases. A final chapter discusses the notion of quantum entanglement and certain fundamental aspects of quantum mechanics, in particular the Bell inequalities and their violations.

Finally note that we have not treated either the philosophical implications of quantum mechanics, or the diverse interpretations of this theory, despite the great interest of these subjects. We have in fact limited ourselves to presenting what is commonly called the “orthodox point of view”. It is only in Chapter XXI that we touch on certain questions concerning the foundations of quantum mechanics (nonlocality, etc.). We have made this choice because we feel that one can address such questions more efficiently after mastering the manipulation of the quantum mechanical formalism as well as its numerous applications. These subjects are addressed in the book Do we really understand quantum mechanics? (F. Laloë, Cambridge University Press, 2019); see also section 5 of the bibliography of volumes I and II.

Acknowledgments:

Volumes I and II:

The teaching experience out of which this text grew were group efforts, pursued over several years. We wish to thank all the members of the various groups and particularly Jacques Dupont-Roc and Serge Haroche, for their friendly collaboration, for the fruitful discussions we have had in our weekly meetings and for the ideas for problems and exercises that they have suggested. Without their enthusiasm and valuable help, we would never have been able to undertake and carry out the writing of this book.

Nor can we forget what we owe to the physicists who introduced us to research, Alfred Kastler and Jean Brossel for two of us and Maurice Levy for the third. It was in the context of their laboratories that we discovered the beauty and power of quantum mechanics. Neither have we forgotten the importance to us of the modern physics taught at the C.E.A. by Albert Messiah, Claude Bloch and Anatole Abragam, at a time when graduate studies were not yet incorporated into French university programs.

We wish to express our gratitude to Ms. Aucher, Baudrit, Boy, Brodschi, Emo, Heywaerts, Lemirre, Touzeau for preparation of the mansucript.

Volume III:

We are very grateful to Nicole and Daniel Ostrowsky, who, as they translated this Volume from French into English, proposed numerous improvements and clarifications. More recently, Carsten Henkel also made many useful suggestions during his translation of the text into German; we are very grateful for the improvements of the text that resulted from this exchange. There are actually many colleagues and friends who greatly contributed, each in his own way, to finalizing this book. All their complementary remarks and suggestions have been very helpful and we are in particular thankful to:

Pierre-François Cohadon

Jean Dalibard

Sébastien Gleyzes

Markus Holzmann

Thibaut Jacqmin

Philippe Jacquier

Amaury Mouchet

Jean-Michel Raimond

Félix Werner

Some delicate aspects of Latex typography have been resolved thanks to Marco Picco, Pierre Cladé and Jean Hare. Roger Balian, Edouard Brézin and William Mullin have offered useful advice and suggestions. Finally, our sincere thanks go to Geneviève Tastevin, Pierre-François Cohadon and Samuel Deléglise for their help with a number of figures.